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On a Special Class of Fibrations and Kähler Rigidity Pacific Journal of Mathematics ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY NICKOLAS J. MICHELACAKIS Volume 219 No. 2 April 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 219, No. 2, 2005 ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY NICKOLAS J. MICHELACAKIS Let ᏭᏮn be the class of torsion-free, discrete groups that contain a normal, at most n-step, nilpotent subgroup of finite index. We give sufficient condi- tions for the fundamental group of a fibration F → T → B, with base B an infra-nilmanifold, to belong to ᏭᏮn. Manifolds of this kind may, for exam- ple, appear as thin ends of nonpositively curved manifolds. We prove that if, in addition, we require that T be Kähler, then T possesses a flat Riemannian metric and the fundamental group π1(T) is necessarily a Bieberbach group. Further, we prove that a torsion-free, virtually polycyclic group that can be realised as the fundamental group of a compact, Kähler K(π, 1)-manifold is necessarily Bieberbach. 1. Introduction Torsion-free, discrete, cocompact subgroups of the group of affine motions of ޒn were first studied by Bieberbach in 1912, and more recently by Charlap; they are called Bieberbach groups. They correspond precisely to the fundamental groups of compact manifolds endowed with a flat Riemannian metric [Charlap 1965], and such manifolds are finitely covered by flat tori [Bieberbach 1911]. L. Auslander [1960] and Lee and Raymond [1985] turned their attention to almost-Bieberbach groups, that is, torsion-free, discrete, cocompact subgroups of GoC, with C a maximal, compact subgroup of Aut G for G a simply con- nected, nilpotent Lie group. They succeeded in generalising much of Bieberbach’s work. Malcev’s equivalence [1949] shows that torsion-free, finitely generated, nilpotent groups correspond precisely to the fundamental groups of nilmanifolds, that is, compact manifolds of the form M = G/N, where G is a simply con- nected, nilpotent Lie group, and N a discrete subgroup. Theorem 3.2 shows that almost-Bieberbach groups correspond to infra-nilmanifolds, compact manifolds of the form G/ 0 with G as above and 0 a discrete subgroup of GoC, where MSC2000: primary 22E40; secondary 32Q15, 14R20. Keywords: affinely flat manifold, (almost)-crystallographic, (almost)-Bieberbach group, (almost)-torsion-free, (virtually) polycyclic group, nilpotent Lie group, discrete cocompact subgroups, lattice, Malcev completion, cohomology of groups, complex (Kähler) structure, group action, group representation, flat Riemannian manifold, (infra)-nilmanifold. 311 312 NICKOLASJ.MICHELACAKIS C is a maximal compact subgroup of Aut G. We denote by ᏭᏮn the class of almost-Bieberbach groups whose maximal, normal, nilpotent subgroup is at most n-step nilpotent. We shall say that a group 0 admits an n-step almost-Bieberbach structure if and only if 0 ∈ ᏭᏮn and its maximal normal nilpotent subgroup is n-step nilpotent. (We know from [Gromov 1981] and [Wolf 1968] that, among finitely generated groups, virtually nilpotent groups are precisely those groups that have polynomial growth. For details and precise definitions, see those works or [Tits 1981].) We employ algebraic methods to study closed manifolds that fibre over infra- nilmanifolds. If F → T → B is such a fibration, where F, T and B are all acyclic, the long homotopy exact sequence reduces to a group extension of the form 1 −→ π1(F) −→ π1(T ) −→ π1(B) −→ 1. Manifolds of this type appear as thin ends of geometrically finite hyperbolic man- ifolds, which are an interesting subclass of nonpositively curved manifolds. More specifically, Apanasov and Xie [1997] proved that if 0 ⊂ ᏴnoU(n−1) is a torsion- n−1 free discrete group acting on the Heisenberg group Ᏼn := ރ ×ޒ, the orbit space Ᏼn/ 0 is a Heisenberg manifold of zero Euler characteristic and a vector bundle over a compact manifold. Further, this compact manifold is finitely covered by a nilmanifold which is either a torus or a torus bundle over a torus. This gener- alises earlier results on almost flat manifolds concerning lattices in ᏴnoU(n − 1) [Gromov 1978; Buser and Karcher 1981]. As mentioned above, groups in ᏭᏮn correspond to infra-nilmanifolds. In Sec- tion 2 we study extensions of the form 1 −→ G −→ 0 −→ K −→ 1, with K ∈ ᏭᏮn, to provide sufficient conditions under which 0 belongs to ᏭᏮn. In particular, Proposition 2.2 guarantees the existence of an almost-Bieberbach structure on 0 provided G is a normal subgroup of 0 in a precise way. Proposition 2.4 does the same provided G lies in ᏭᏮn and the action of K on G respects some suitable minimal conditions. In Section 3 we use the Johnson–Rees characterisation of fundamental groups of flat, Kähler [Johnson and Rees 1991], and projective [Johnson 1990] mani- folds, and apply the Benson–Gordon theorem [1988] for the existence of a Kähler structure on a compact nilmanifold to show, in Theorem 3.3, that the existence of a Kähler structure on a special fibration as above implies the existence of a flat Riemann metric on T . In particular, in ᏭᏮn, the classes of fundamental groups of Kähler and projective manifolds coincide, as shown in Corollary 3.4. Further, as a consequence of the Lefschetz hyperplane theorem and Bertini’s theorem, this ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY 313 is a subclass of the class of fundamental groups of compact, closed, nonsingular projective surfaces. Finally, in Section 4, we use a structure theorem concerning virtually polycyclic groups, proved in [Dekimpe and Igodt 1994], together with the results in [Arapura and Nori 1999], to prove, in Theorem 4.2, that a torsion-free, virtually polycyclic group can be realised as the fundamental group of a K(π, 1)-compact, Kähler manifold if and only if it is Bieberbach of a special kind, namely, its operator homomorphism is essentially complex. 2. Group extensions A group N is said to be nilpotent if its upper central series 1 = N0 C N1 = Z(N) C N2 C ··· , defined by Ni+1/Ni = Z(N/Ni ), is finite. If n is the smallest integer such that Nn = N, then N is said to be n-step nilpotent. We shall say that a finitely generated, torsion-free group 0 admits an (n-step) almost-Bieberbach group structure if it can be written as an extension of a finitely generated, (n-step) nilpotent group N by a finite group 8. Notice that, given such a torsion-free, finitely generated, nilpotent ∼ i group, its quotients Ni+1/Ni are of a special form, namely Ni+1/Ni = ޚ j . Lemma 2.1. Let 0 fit in an extension p 0 −→ ޚm −→ 0 −→ G −→ 1, where the torsion-free group G has an n-step nilpotent, normal subgroup N of finite index and ޚm a trivial N-module. Then 0 ∈ ᏭᏮn+1. Proof. Let G be defined by the extension 1 −→ N −→ G −→ 8 −→ 1, with N n-step nilpotent, 8 finite and φ : 8 → Out N the operator homomorphism. Consider 0 := p−1(N). Then the extension p (2–1) 0 −→ ޚm −→ 0 −→ N −→ 1 is central, which implies that 0 is at most (n+1)-step nilpotent. The proof is −1 ∼ m m ∼ completed by the observation that 0 = p (N)C0 and 0/0 = (0/ޚ )/(0/ޚ ) = ∼ m G/N = 8. Notice that 0 is torsion-free since so are ޚ and G. We now turn our attention to the fibre of the fibration F → T → B to prove the following: 314 NICKOLASJ.MICHELACAKIS Proposition 2.2. Let 0 be a torsion-free extension p (2–2) 1 −→ K −→ 0 −→ G −→ 1 of a finitely generated group K by a group G admitting an n-step nilpotent, almost- Bieberbach structure such that Im c is finite, where c : 0 → Aut K denotes the conjugation map. Then 0 ∈ ᏭᏮn+1. Proof. Since G admits an n-step almost-Bieberbach structure there is a short exact sequence 1 −→ N −→ G −→ 8 −→ 1 where N is n-step nilpotent, 8 is finite, and φ : 8 → Out N is the operator homo- morphism. Let 0ˆ := p−1(N). Then 0ˆ fits in a short exact sequence p 1 −→ K −→ 0ˆ −→ N −→ 1, where we denote by c¯ the restriction of the conjugation map c : 0 → Aut K to 0ˆ . Let 0 := Ker c¯, which is nonempty since 0 is infinite. Then the extension 1 −→ 0 ∩ K −→ 0 −→ p(0) −→ 1 is central, with p(0) C N, and therefore itself nilpotent. This means that 0 ∩ K is a finitely generated, torsion-free, abelian group and 0 is at most (n+1)-step nilpotent. The proof is completed by observing that the normal subgroup 0 of 0ˆ ˆ has finite index in 0, since Im c¯ is finite. The group Aut K , for K a Bieberbach group, is not necessarily finite. For an example, see [Charlap 1986, p. 219]. It does, then, make sense to check what happens if the fibre admits an n-step almost-Bieberbach structure. But first: Proposition 2.3. Let 0 be a torsion-free extension p 1 −→ K −→ 0 −→ ޚn −→ 0 of a Bieberbach group K by a free abelian group of rank n, such that ޚm ⊆ Z(0), where ޚm is the translation subgroup of K and Z(0) the center of 0. Then 0 ∈ ᏭᏮ2. m m Proof. First observe that ޚ C 0, since ޚ ≤ Z(0). We therefore have a short exact sequence p (2–3) 1 −→ K/ޚm −→ 0/ޚm −→ ޚn −→ 0 where K/ޚm is isomorphic to F, the finite holonomy group of K . We distinguish two cases: ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY 315 n−1 n (i) Assume that (2–3) is a central extension.
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